Notes on Thermodynamics

The following sections are included:

• Preface
• Contents
• List of Examples

The following sections are included:

• Heat, Mechanics, Thermodynamics
• System, surroundings, universe
• The PVT equation of state
• Internal energy
• Internal energy of ideal gases
• Heat capacity

The following sections are included:

• Thermodynamic state, intensive and extensive variables
• Equilibrium
• What is Temperature?
• The Joule experiment
• The First Law
• Other forms of work
• Reversible and irreversible processes
• Properties of the internal energy
• The internal energy of an ideal gas

We discuss some thermodynamics ideas that follow from the First Law. We begin with some typical “problems” making use of the ideal gas as a simple model substance to make the discussion concrete. Then we will introduce the concept of Enthalpy and use that to discuss thermochemistry applications, the throttling of real gases and intermolecular interactions.

If all human interactions were simply driven by a relentless natural increase of, say, net happiness, and if we knew how to calculate the amount of happiness in any situation, we would be able to predict the course of any human interaction. Such an encompassing law of human interaction would be on par with the Second Law of Thermodynamics for understanding Nature. And happiness would just be the analogue in human affairs of the quantity we call entropy in describing Nature. We will take this chapter and the next to get acquainted with the idea of entropy and this most powerful law of Nature. To begin with, we discuss the Carnot cycle using the skills acquired from the previous chapters in calculating heat and work interactions. The Carnot cycle, an idealized cyclic process of compression and expansion of a gas, was conceived in the early 1800’s by an engineer Sadi Carnot who built on his father’s work to analyse the efficiency of steam engines. That rather practical goal led Carnot to the Second Law of Thermodynamics. His analysis subsequently led Clausius to the concept of entropy. Indeed, Carnot’s analysis of the steam engine was arguably the start of Thermodynamics; the Second Law was the first amongst the laws of Thermodynamics to be formulated.

Even though we examined the efficiency of engines, you would have noticed that our discussion in the last chapter was most general, in terms only of the heat energy supplied to and lost by an engine, and the work extracted from it. No engineering detail of any machinery was examined. That was sufficient, and preferred, because we are trying to get to the fundamental law that governs heat and work interactions in general, rather than just the effectiveness of the specific construction of any engine. It was Carnot’s genius to recognize that behind the problem of the efficiency of real steam engines there lies a fundamental principle that governs all of Nature. Although heat and work are both forms of energy, you will realise that heat is different from the other forms of energy. Work always needs to be done to pump heat from colder objects to hotter ones. In the reverse direction, a hot object can lose heat to a cold object without requiring any work to be done. No engine can ever completely convert an amount of heat to work. The other way around, you can always completely dissipate work into heat. There is an asymmetry in the direction of natural phenomena involving Heat. Our task in this chapter is to identify the one quantity that allows us to characterise the spontaneous direction of any process in Nature.

By relating entropy to heat and temperature in reversible processes, the Second Law of Thermodynamics provides a means of calculating the entropy change of a system transforming from one state to another. From this, we have:

The following sections are included:

• The four Laws of Thermodynamics
• Criteria for spontaneous processes
• What is the maximum work that can be extracted from any system?
• The Gibbs free energy of formation
• Thermodynamic potentials H, A, G are extensive variables
• The fundamental equations
• Using the fundamental equations
  • Maxwell relations
  • Applying the Maxwell relations
• Natural variables
• (∂U/∂V)_T, (∂H/∂P)_T and intermolecular interactions
• How the Gibbs free energy depends on T and P
• From the Gibbs free energy G(T, P) to thermodynamic properties
• The Gibbs-Helmholtz equation
• ΔG and ΔH at low temperatures

The following sections are included:

• Scaling of Gibbs free energy with system size
• μ is the molar Gibbs free energy
• Chemical potentials in multicomponent systems G = Σ_iμ_in_i
• The fundamental equations for an open system
• Partial molar quantities
• Equilibrium condition: minimizing G in a multiphase system
• The Gibbs Phase Rule
• The Gibbs-Duhem equation
  • How Gibbs derived his Phase Rule
  • The Gibbs-Duhem equation for other partial molar quantities
• Phase diagrams for single-component systems
• The Clapeyron equation: equilibrium between two phases
• Using the Clapeyron equation
  • The Clausius-Clapeyron equation
  • Temperature-dependent Δ_vapH
  • A non-ideal vapour phase

In learning a new physical theory, it is always easier to have accessible concrete models to apply the theory to. We offer some such models in decreasing levels of familiarity in this chapter. Knowing the Gibbs free energy as a function of temperature and pressure provides all thermodynamic information. Thus, our approach is to get to expressions for the chemical potential of various systems. In Chapter 1 we introduced the ideal gas and the van der Waals gas as model substances used to illustrate the calculation of thermodynamic properties. We have used the equations of state of these substances in various examples along the way. At this point we have a more sophisticated framework for thermodynamics than can be usefully illustrated with only the ideal gas and the van der Waals gas. Therefore, we apply this framework to a few model substances/systems which are slightly more sophisticated than the ideal gas. These model systems will be used in further applications in later chapters.

The ideal gas is deficient as a physical model because no matter how low the temperature and/or how high the pressure, it does not condense to form a liquid phase. This is because the intermolecular interactions are taken to be zero. In the previous chapter, we have introduced the chemical potentials of non-ideal gases and solutions which include the influence of intermolecular interactions. Using these chemical potentials, we discussed how to calculate, for example, the fugacity of non-ideal gases using the van der Waals equation of state. Here, we expand upon that by providing a more detailed discussion of the van der Waals fluid because this is an important elementary model for understanding the phase transformation from gas to liquid. In Section 7.10, we looked at the gas-liquid phase transformation using one equation to describe each of the two phases. Historically, the van der Waals equation of state was the first to describe this phase transformation using the same equation of state for both the gas and liquid phases. It is not accurate quantitatively as a description of specific substances, especially in their liquid states, but it illustrates the essential role of intermolecular interactions. In this chapter, we further examine this equation of state, starting with its description of phase change.

The following sections are included:

• Colligative properties of dilute solutions
  • Lowering of the freezing point
  • Elevation of the boiling point
  • The osmotic pressure
• Solution-pure solid equilibrium
• From tie-line to the lever rule
• Binary gas-liquid equilibrium
• Binary gas-liquid equilibrium: T-xy diagram
• Distillation
• Mixing in non-ideal systems

In your chemistry lab in school, you must have precipitated calcium carbonate using the reaction between aqueous solutions of sodium carbonate and calcium chloride:

The following sections are included:

• The electrochemical cell
• From Δ_rxn G to φ
• The half-cell potential
• The electrochemical series
• The electrochemical cell: notation
• Nernst equation
• The temperature dependence of φ^∅

The following section is included:

• Index